Optical Experimental Evidence for a Universal Length Scale for the Dynamic Charge Inhomogeneity of Cuprate Superconductors

Time-resolved optical experiments can give unique information on the characteristic length scales of dynamic charge inhomogeneity on femtosecond time scales. From data on the effective quasiparticle relaxation time ${\tau }_r$ in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ and ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$, we derive the temperature and doping dependence of the intrinsic phonon escape length $l_e$, which can be a direct measure of charge inhomogeneity. Remarkably, a common feature of both $p$- and $n$-type cuprates is that, as $T\to T_c$, $l_e$approaches the superconducting coherence length $l_e\to {\xi }_s(0)$. In the normal state $l_e$ is found to be in excellent agreement with the mean free path $l_m$ obtained from the resistivity data and structural coherence lengths $l_s$ from neutron scattering experiments, implying the existence of complex intrinsic textures on different length scales which may have a profound effect on the functional properties of these materials.

Figure 1

The photoinduced reflectivity $\Delta R/R$ measured on ${\mathrm{La}}_{1.9}{\mathrm{Sr}}_{0.1}{\mathrm{CuO}}_4$ as a function of time for different $T$. The relaxation is described reasonably well by a single exponential. Inset: a schematic representation of the phonon escape process arising from the pairing of two particles with $E_1,k_1$ and $E_2,k_2$ (solid lines) with the emission of acoustic phonons (wavy line) in an inhomogeneous medium. The overall rate of recombination is dominated by phonon escape out of the charged volume $V$.

Figure 2

The phonon escape length $l_e$ as a function of temperature for ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ for $x=0.06$, 0.1, 0.115, 0.15, and 0.2, respectively. The $T_c$ are 0, 31, 32, 38, 5, and 30 K, respectively. ($x=0.06$ is nonsuperconducting.) Here we have used $v_s=4.8\mathrm{nm}{\mathrm{ps}}^{-1}$ for all $x$ in LSCO [19].

Figure 3

(a) A plot of $l_e$ (solid color symbols), $l_m$ (open symbols), and $l_s$ (black symbols), as a function of $T/T_c$ for LSCO at different doping levels. In all cases, the length scale is normalized to ${\xi }_s(0)$ [29]. The inset shows the critical behavior just above $T_c$ for $x=0.1$. The mean-field prediction is plotted $l/{\xi }_{\mathrm{GL}}(T)\sim (T-T_c{)}^{-v}$, with $\nu =1/2$ as a dashed line. (b) The normalized escape lengths $l_e/{\xi }_s(0)$ for ${\mathrm{La}}_{1.85}{\mathrm{Sr}}_{0.15}{\mathrm{CuO}}_4$ (solid squares) and ${\mathrm{Nd}}_{1.85}{\mathrm{Ce}}_{0.15}{\mathrm{CuO}}_4$ (open circles) as a function of $T/T_c$. Here ${\xi }_s(0)=1.8$ and 18 nm, respectively, for LSCO and NCCO [24]. $v_s=5.5\mathrm{nm}{\mathrm{ps}}^{-1}$ for NCCO [30].